On the absolute continuity of infinite product measure and its convolution (Q1117581)

Let \(X=\{X_ k\}\) be an i.i.d. random sequence and \(Y=\{Y_ k\}\) be a symmetric independent random sequence which is also independent of X. Then X and \(X+Y=\{X_ k+Y_ k\}\) induce probability measures \(\mu_ X\) and \(\mu_{X+Y}\) on the sequence space, respectively. The problem is to characterize the absolute continuity of \(\mu_ X\) and \(\mu_{X+Y}\) and give applications to the absolute continuity of stochastic processes; in particular we give a sufficient condition for the absolute continuity of the sum of Brownian motion and an independent process with respect to the Brownian motion. We assume that the distribution of \(X_ 1\) is equivalent to the Lebesgue measure and the density function f satisfies \[ (C)\quad \int^{+\infty}_{-\infty}(f''(x)^ 2/f(x))dx<+\infty. \] Under this condition we shall give some sufficient conditions and necessary conditions for \(\mu_ X\sim \mu_{X+Y}\). The critical condition is \[ \sum_{k}{\mathbb{E}}[| Y_ k|^ 2:| Y_ k| \leq \epsilon]^ 2<+\infty \] for some \(\epsilon >0\). In particular in the case where X is Gaussian, we shall give finer results. Finally we shall compare the condition (C) with the Shepp's condition: \[ (A)\quad \int^{+\infty}_{-\infty}(f'(x)^ 2/f(x))dx<+\infty. \]